flat function

{{short description|Function whose all derivatives vanish at a point}}

File:FBN exp(-1x2).jpeg

In mathematics, especially real analysis, a real function is flat at $x\_0$ if all its derivatives at $x\_0$ exist and equal {{math|0}}.

A function that is flat at $x\_0$ is not analytic at $x\_0$ unless it is constant in a neighbourhood of $x\_0$ (since an analytic function must equals the sum of its Taylor series).

An example of a flat function at {{math|0}} is the function such that $f(0)=0$ and $f(x)=e^\{-1/x^2\}$ for $x\backslash neq\; 0.$

The function need not be flat at just one point. Trivially, constant functions on $\backslash mathbb\{R\}$ are flat everywhere. But there are also other, less trivial, examples; for example, the function such that $f(x)=0$ for $x\backslash leq\; 0$ and $f(x)=e^\{-1/x^2\}$ for $x>\; 0.$